This renewal grant proposes the development and evaluation of improved statistical methods for the analysis of 'failure' time data. Primary areas of application include clinical trials and animal follow-up studies though some topics also pertain to intervention studies and epidemiologic cohort studies. The Cox regression model provides the methodological starting point for much of the proposed work. This model specifies a failure rate function for a study subject that is the product of an unrestricted 'baseline' failure rate function and a parametric relative risk function. The relative risk at a specified follow-up time can depend not only on prestudy treatment assignment and other factors but also on time-dependent covariates obtained during study follow-up. As such the Cox regression method provides a powerful approach to explaining treatment differences or to gaining insight into disease mechanisms. We propose the further development of the theory and application of the Cox regression method by considering the following sub-topics: choice of relative risk form; tests of fit; exploratory data analysis methods and computation time reduction; use of time-dependent covariates; and application of asymptotic distribution theory with moderate sized samples. A second proposed area of emphasis is multivariate failure time methods. Sub-topics include relative risk estimation with renewal-type models and the non-parametric estimation and display of bivariate survivor functions. An important new area of emphasis concerns the methodology for various types of multiple testing problems. Outstanding topics in sequential testing, in prognostic factor identification methods and in the study of treatment effects within subgroups, will be addressed. Topics in the analysis of animal carcinogenesis experiments, in competing risk methodology will also be considered, as will a range of other failure time data topics.